Phạm Văn Thuận: The Mathscope
the mathscope All the best from Vietnamese Problem Solving Journals Updated November 2, 2005 translated by Pham Van Thuan, Eckard Specht Vol I, Problems in Mathematics Journal for the Youth Mathscope is a free problem resource selected from problem solving journals in Vietnam. This freely accessible collection is our effort to introduce elementary mathematics problems to our foreign friends for either recreational or professional use. We would like to give you a new taste of Vietnamese mathematical culture. Whatever the purpose, we welcome suggestions and comments from you all. More communications can be addressed to Pham Van Thuan, 4E2, 565 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam, or email us at pvthuan@vnu.edu.vn. It’s now not too hard to find problems and solutions on the Internet due to the increasing numbers of websites devoted to mathematical problems solving. Anyway, we hope that this complete collection saves you considerable time searching the problems you really want. We intend to give an outline of solutions to the problems, but it would take time. Now enjoy these “cakes” from Vietnam first. 261. 1 (Ho Quang Vinh) Given a triangle ABC, its internal angle bi- sectors BE and CF , and let M be any point on the line segment EF . De- note by S A , S B , and S C the areas of triangles MBC, MCA, and MAB, respectively. Prove that √ S B + √ S C √ S A ≤ AC + AB BC , and determine when equality holds. 1 All the best from Vietnamese Problem Solving Journals 261. 2 (Editorial Board) Find the maximum value of the expression A = 13 x 2 − x 4 + 9 x 2 + x 4 for 0 ≤ x ≤ 1. 261. 3 (Editorial Board) The sequence (a n ), n = 1, 2, 3, . . . , is defined by a 1 > 0, and a n+1 = ca 2 n + a n for n = 1, 2, 3, . . . , where c is a constant. Prove that a) a n ≥ c n−1 n n a n+1 1 , and b) a 1 + a 2 + ··· + a n > n na 1 − 1 c for n ∈ N. 261. 4 (Editorial Board) Let X, Y , Z be the reflections of A, B, and C across the lines BC, CA, and AB, respectively. Prove that X, Y , and Z are collinear if and only if cos A cos B cos C = − 3 8 . 261. 5 (Vinh Competition) Prove that if x, y, z > 0 and 1 x + 1 y + 1 z = 1 then the following inequality holds: 1 − 1 1 + x 2 1 − 1 1 + y 2 1 − 1 1 + z 2 > 1 2 . 261. 6 (Do Van Duc) Given four real numb e rs x 1 , x 2 , x 3 , x 4 such that x 1 + x 2 + x 3 + x 4 = 0 and |x 1 |+ |x 2 |+ |x 3 |+ |x 4 | = 1, find the maximum value of 1≤i<j≤4 (x i − x j ). 261. 7 (Doan Quang Manh) Given a rational number x ≥ 1 such that there exists a sequence of integers (a n ), n = 0, 1, 2, . . . , and a constant c = 0 such that lim n→∞ (cx n − a n ) = 0. Prove that x is an integer. 262. 1 (Ngo Van Hiep) Let ABC an equilateral triangle of side length a. For each point M in the interior of the triangle, choose points D, E, F on the sides CA, AB, and BC, respectively, such that DE = MA, EF = MB, and F D = MC. Determine M such that △DEF has smallest possible area and calculate this area in terms of a. 262. 2 (Nguyen Xuan Hung) Given is an acute triangle with altitude AH. Let D be any point on the line segment AH not coinciding with the endpoints of this segment and the orthocenter of triangle ABC. Let ray BD intersect AC at M, ray CD meet AB at N. The line perpendicular to BM at M meets the line perpendicular to CN at N in the point S. Prove that △ABC is isosceles with base BC if and only if S is on line AH. 2 All the best from Vietnamese Problem Solving Journals 262. 3 (Nguyen Duy Lien) The sequence (a n ) is defined by a 0 = 2, a n+1 = 4a n + 15a 2 n − 60 for n ∈ N. Find the general term a n . Prove that 1 5 (a 2n + 8) can be expressed as the sum of squares of three consecutive integers for n ≥ 1. 262. 4 (Tuan Anh) Let p be a prime, n and k positive integers with k > 1. Suppose that b i , i = 1, 2, . . . , k, are integers such that i) 0 ≤ b i ≤ k − 1 for all i, ii) p nk−1 is a divisor of k i=1 p nb i −p n(k−1) − p n(k−2) − ··· −p n − 1. Prove that the sequence (b 1 , b 2 , . . . , b k ) is a permutation of the sequence (0, 1, . . . , k −1). 262. 5 (Doan The Phiet) Without use of any calculator, determine sin π 14 + 6 sin 2 π 14 − 8 sin 4 π 14 . 264. 1 (Tran Duy Hinh) Prove that the sum of all squares of the divi- sors of a natural number n is less than n 2 √ n. 264. 2 (Hoang Ngoc Canh) Given two p oly nomials f(x) = x 4 − (1 + e x ) + e 2 , g(x) = x 4 − 1, prove that for distinct positive numb e rs a, b satisfying a b = b a , we have f(a)f(b) < 0 and g(a)g(b) > 0. 264. 3 (Nguyen Phu Yen) Solve the equation (x −1) 4 (x 2 − 3) 2 + (x 2 − 3) 4 + 1 (x −1) 2 = 3x 2 − 2x −5. 264. 4 (Nguyen Minh Phuong, Nguyen Xuan Hung) Let I be the incenter of triangle ABC. Rays AI, BI, and CI meet the circumcircle of triangle ABC again at X, Y , and Z, respectively. Prove that a) IX + IY + IZ ≥ IA + IB + IC, b) 1 IX + 1 IY + 1 IZ ≥ 3 R . 265. 1 (Vu Dinh Hoa) The lengths of the four sides of a convex quadri- lateral are natural numbers such that the sum of any three of them is di- visible by the fourth number. Prove that the quadrilateral has two equal sides. 3 All the best from Vietnamese Problem Solving Journals 265. 2 (Dam Van Nhi) Let AD, BE, and CF be the internal angle bi- sectors of triangle ABC. Prove that p(DEF ) ≤ 1 2 p(ABC), where p(XY Z) denotes the perimeter of triangle XY Z. When does equality hold? 266. 1 (Le Quang Nam) Given real numbers x, y , z ≥ −1 satisfying x 3 + y 3 + z 3 ≥ x 2 + y 2 + z 2 , prove that x 5 + y 5 + z 5 ≥ x 2 + y 2 + z 2 . 266. 2 (Dang Nhon) Let ABCD be a rhombus with ∠A = 120 ◦ . A ray Ax and AB make an angle of 15 ◦ , and Ax meets BC and CD at M and N, respectively. Prove that 3 AM 2 + 3 AN 2 = 4 AB 2 . 266. 3 (Ha Duy Hung) Given an isosceles triangle with ∠A = 90 ◦ . Let M be a variable point on line BC, (M distinct from B, C). Let H and K be the orthogonal projections of M onto lines AB and AC, respectively. Suppose that I is the intersection of lines CH and BK. Prove that the line MI has a fixed point. 266. 4 (Luu Xuan Tinh) Let x, y be real numbers in the interval (0, 1) and x + y = 1, find the minimum of the expression x x + y y . 267. 1 (Do Thanh Han) Let x, y, z be real numbers such that x 2 + z 2 = 1, y 2 + 2y(x + z) = 6. Prove that y(z − x) ≤ 4, and determine when equality holds. 267. 2 (Le Quoc Han) In triangle ABC, medians AM and CN meet at G. Prove that the quadrilateral BMGN has an incircle if and only if triangle ABC is isosceles at B. 267. 3 (Tran Nam Dung) In triangle ABC, denote by a, b, c the side lengths, and F the area. Prove that F ≤ 1 16 (3a 2 + 2b 2 + 2c 2 ), and determine when equality holds. Can we find another set of the coef- ficients of a 2 , b 2 , and c 2 for which equality holds? 268. 1 (Do Kim Son) In a triangle, denote by a, b, c the side lengths, and let r, R be the inradius and circumradius, respectively. Prove that a(b + c −a) 2 + b(c + a −b) 2 + c(a + b −c) 2 ≤ 6 √ 3R 2 (2R −r). 4 All the best from Vietnamese Problem Solving Journals 268. 2 (Dang Hung Thang) The sequence (a n ), n ∈ N, is defined by a 0 = a, a 1 = b, a n+2 = da n+1 − a n for n = 0, 1, 2, . . . , where a, b are non-zero integers, d is a r eal number. Find all d such that a n is an integer for n = 0, 1, 2, . . . . 271. 1 (Doan The Phiet) Find necessary and sufficient conditions with respect to m such that the system of equations x 2 + y 2 + z 2 + xy − yz − zx = 1, y 2 + z 2 + yz = 2, z 2 + x 2 + zx = m has a solution. 272. 1 (Nguyen Xuan Hung) Given are three externally tangent cir- cles (O 1 ), (O 2 ), and (O 3 ). Let A, B, C be resp e ctively the points of tan- gency of (O 1 ) and (O 3 ), (O 2 ) and (O 3 ), (O 1 ) and (O 2 ). The common tangent of (O 1 ) and (O 2 ) meets C and (O 3 ) at M and N. Let D be the midpoint of MN. Prove that C is the center of one of the excircles of triangle ABD. 272. 2 (Trinh Bang Giang) Let ABCD b e a convex quadrilateral such that AB + CD = BC + DA. Find the locus of points M interior to quadrilateral ABCD such that the sum of the distances from M to AB and CD is equal to the sum of the distances from M to BC and DA. 272. 3 (Ho Quang Vinh) Let M and m be the greatest and smallest numbers in the set of positive numbers a 1 , a 2 , . . . , a n , n ≥ 2. Prove that n i=1 a i n i=1 1 a i ≤ n 2 + n(n −1) 2 M m − m M 2 . 272. 4 (Nguyen Huu Du) Find all primes p such that f(p) = (2 + 3) −(2 2 + 3 2 ) + (2 3 + 3 3 ) −··· −(2 p−1 + 3 p−1 ) + (2 p + 3 p ) is divisible by 5. 274. 1 (Dao Manh Thang) Let p be the semiperimeter and R the cir- cumradius of triangle ABC. Furthermore, let D, E, F be the excenters. Prove that DE 2 + EF 2 + FD 2 ≥ 8 √ 3pR, and determine the equality case. 5 All the best from Vietnamese Problem Solving Journals 274. 2 (Doan The Phiet) Detemine the positive root of the equation x ln 1 + 1 x 1+ 1 x −x 3 ln 1 + 1 x 2 1+ 1 x 2 = 1 −x. 274. 3 (N.Khanh Nguyen) Let ABCD be a cyclic quadrilateral. Points M, N, P , and Q are chosen on the sides AB, BC, CD, and DA, re- spectively, such that MA/M B = P D/P C = AD/BC and QA/QD = NB/NC = AB/CD . Prove that MP is perpendicular to NQ. 274. 4 (Nguyen Hao Lieu) Prove the inequality for x ∈ R: 1 + 2x arctan x 2 + ln(1 + x 2 ) 2 ≥ 1 + e x 2 3 + e x . 275. 1 (Tran Hong Son) Let x, y, z be real numbers in the interval [−2, 2], prove the inequality 2(x 6 + y 6 + z 6 ) −(x 4 y 2 + y 4 z 2 + z 4 x 2 ) ≤ 192. 276. 1 (Vu Duc Canh) Find the maximum value of the expression f = a 3 + b 3 + c 3 abc , where a, b, c are real numbers lying in the interval [1, 2]. 276. 2 (Ho Quang Vinh) Given a triangle ABC with sides BC = a, CA = b, and AB = c. Let R and r be the circumradius and inradius of the triangle, respectively. Prove that a 3 + b 3 + c 3 abc ≥ 4 − 2r R . 276. 3 (Pham Hoang Ha) Given a triangle ABC, let P be a point on the side BC, let H, K be the orthogonal projections of P onto AB, AC respectively. Points M, N are chosen on AB, AC such that PM AC and P N AB. Compare the areas of triangles P HK and PMN. 276. 4 (Do Thanh Han) How many 6-digit natural numbers exist with the distinct digits and two arbitrary consecutive digits can not be simul- taneously odd numbers? 277. 1 (Nguyen Hoi) The incircle with center O of a triangle touches the sides AB, AC, and BC respectively at D, E, and F. The escribed circle of triangle ABC in the angle A has center Q and touches the side BC and the rays AB, AC respectively at K, H, and I. The line DE meets the rays BO and CO respectively at M and N. The line HI meets the rays BQ and CQ at R and S, respectively. Prove that a) △F M N = △KRS, b) IS AB = SR BC = RH CA . 6 All the best from Vietnamese Problem Solving Journals 277. 2 (Nguyen Duc Huy) Find all rational numbers p, q, r such that p cos π 7 + q cos 2π 7 + r cos 3π 7 = 1. 277. 3 (Nguyen Xuan Hung) Let ABCD be a bicentric quadrilateral inscribed in a circle with center I and circumcribed about a circle with center O. A line through I, parallel to a side of ABCD, intersects its two opposite sides at M and N. Prove that the length of M N does not depend on the choice of side to which the line is parallel. 277. 4 (Dinh Thanh Trung) Let x ∈ (0, π) be real number and sup- pose that x π is not rational. Define S 1 = sin x, S 2 = sin x + sin 2x, . . . , S n = sin x + sin 2x + ··· + sin nx. Let t n be the number of negative terms in the sequence S 1 , S 2 , . . . , S n . Prove that lim n→∞ t n n = x 2π . 279. 1 (Nguyen Huu Bang) Find all natural numbers a > 1, such that if p is a prime divisor of a then the number of all divisors of a which ar e relatively prime to p, is equal to the number of the divisors of a that are not relatively prime to p. 279. 2 (Le Duy Ninh) Prove that for all real numbers a, b, x, y satisfy- ing x + y = a + b and x 4 + y 4 = a 4 + b 4 then x n + y n = a n + b n for all n ∈ N. 279. 3 (Nguyen Huu Phuoc) Given an equilateral triangle ABC, find the locus of p oints M interior to ABC such that if the orthogonal pro- jections of M onto BC, CA and AB are D, E, and F , respectively, then AD, BE, and CF are concurrent. 279. 4 (Nguyen Minh Ha) Let M be a poi nt in the interior of triangle ABC and let X, Y , Z be the reflections of M across the sides BC, CA, and AB, respectively. Prove that triangles ABC and XY Z have the same centroid. 279. 5 (Vu Duc Son) Find all positive integers n such that n < t n , where t n is the number of positive divisors of n 2 . 279. 6 (Tran Nam Dung) Find the maximum value of the expression x 1 + x 2 + y 1 + y 2 + z 1 + z 2 , where x, y, z are real numbers satisfying the condition x + y + z = 1. 7 All the best from Vietnamese Problem Solving Journals 279. 7 (Hoang Hoa Trai) Given are three concentric circles with center O, and radii r 1 = 1, r 2 = √ 2, and r 3 = √ 5. Let A, B, C be three non- collinear points lying resp ectively on these circles and let F be the area of triangle ABC. Prove that F ≤ 3, and determine the side lengths of triangle ABC. 281. 1 (Nguyen Xuan Hung) Let P be a point exterior to a circle with center O. From P construct two tangents touching the circle at A and B . Let Q be a point, distinct from P, on the circle. The tangent at Q of the circle intersects AB and AC at E and F, respectively. Let BC intersect OE and OF at X and Y , respectively. Prove that XY/EF is a constant when P varies on the circle. 281. 2 (Ho Quang Vinh) In a triangle ABC, let BC = a, CA = b, AB = c be the sides, r, r a , r b , and r c be the inradius and exradii. Prove that abc r ≥ a 3 r a + b 3 r b + c 3 r c . 283. 1 (Tran Hong Son) Simplify the expression x(4 −y)(4 −z) + y(4 −z)(4 − x) + z(4 − x)(4 −y) − √ xyz, where x, y, z are posi tive numbers such that x + y + z + √ xyz = 4. 283. 2 (Nguyen Phuoc) Let ABCD be a convex quadrilateral, M be the midpoint of AB. Point P is chosen on the segment AC such that lines MP and BC intersect at T . Suppose that Q is on the segment BD such that BQ/QD = AP/P C. Prove that the li ne TQ has a fixed point when P moves on the segment AC. 284. 1 (Nguyen Huu Bang) Given an integer n > 0 and a prime p > n + 1, prove or disprove that the following equation has integer solutions: 1 + x n + 1 + x 2 2n + 1 + ··· + x p pn + 1 = 0. 284. 2 (Le Quang Nam) Let x, y be real numbers such that (x + 1 + y 2 )(y + 1 + x 2 ) = 1, prove that (x + 1 + x 2 )(y + 1 + y 2 ) = 1. 8 All the best from Vietnamese Problem Solving Journals 284. 3 (Nguyen Xuan Hung) The internal angle bisectors AD, BE, and CF of a triangle ABC meet at point Q. Prove that if the inradii of triangles AQF , BQD, and CQE are equal then triangle ABC is equilat- eral. 284. 4 (Tran Nam Dung) Disprove that there exists a polynomial p(x) of degree greater than 1 such that if p(x) is an integer then p(x + 1) is also an integer for x ∈ R. 285. 1 (Nguyen Duy Lien) Given an odd natural number p and inte- gers a, b, c, d, e such that a + b + c + d + e and a 2 + b 2 + c 2 + d 2 + e 2 are all divisible by p. Prove that a 5 + b 5 + c 5 + d 5 + e 5 − 5abcde is also divisible by p. 285. 2 (Vu Duc Canh) Prove that if x, y ∈ R ∗ then 2x 2 + 3y 2 2x 3 + 3y 3 + 2y 2 + 3x 2 2y 3 + 3x 3 ≤ 4 x + y . 285. 3 (Nguyen Huu Phuoc) Let P be a point in the interior of trian- gle ABC. Rays AP, BP, and CP intersect the sides BC, CA, and AB at D, E, and F, respectively. Let K be the point of i ntersection of DE and CM , H be the point of intersection of DF and BM. Prove that AD, BK and CH are concurrent. 285. 4 (Tran Tuan Anh) Let a, b, c be non-negative real numbers, de- termine all real numbers x such that the following inequality holds: [a 2 + b 2 + (x −1)c 2 ][a 2 + c 2 + (x −1)b 2 ][b 2 + c 2 + (x −1)a 2 ] ≤ (a 2 + xbc)(b 2 + xac)(c 2 + xab). 285. 5 (Truong Cao Dung) Let O and I be the circumcenter and in- center of a triangle ABC. Rays AI, BI, and CI meet the circumcircle at D, E, and F , respectively. Let R a , R b , and R c be the radii of the escribed circles of △ABC, and let R d , R e , and R f be the radii of the escribed circles of triangle DEF . Prove that R a + R b + R c ≤ R d + R e + R f . 285. 6 (Do Quang Duong) Determine all integers k such that the se- quence defined by a 1 = 1, a n+1 = 5a n + ka 2 n − 8 for n = 1, 2, 3, . . . includes only integers. 286. 1 (Tran Hong Son) Solve the equation 18x 2 − 18x √ x −17x −8 √ x −2 = 0. 9 All the best from Vietnamese Problem Solving Journals 286. 2 (Pham Hung) Let ABCD be a square. Points E, F are chosen on CB and CD, respectively, such that BE/BC = k, and DF/DC = (1−k)/(1+k), where k is a given number, 0 < k < 1. Segment BD meets AE and AF at H and G, resp ectively. The line through A, perpendicular to EF , intersects BD at P . Prove that P G/P H = DG/BH. 286. 3 (Vu Dinh Hoa) In a convex hexagon, the se gment joining two of its vertices, dividing the hexagon into two quadrilaterals is called a principal diagonal. Prove that in every convex hexagon, in which the length of each side is equal to 1, there exists a principal diagonal with length not greater than 2 and there exists a principal diagonal with length greater than √ 3. 286. 4 (Do Ba Chu) Prove that in any acute or right triangle ABC the following inequality holds: tan A 2 + tan B 2 + tan C 2 + tan A 2 tan B 2 tan C 2 ≥ 10 √ 3 9 . 286. 5 (Tran Tuan Diep) In triangle ABC, no angle exceeding π 2 , and each angle is greater than π 4 . Prove that cot A + cot B + cot C + 3 cot A cot B cot C ≤ 4(2 − √ 2). 287. 1 (Tran Nam Dung) Suppose that a, b are positive integers such that 2a −1, 2b −1 and a + b are all primes. Prove that a b + b a and a a + b b are not divisible by a + b. 287. 2 (Pham Dinh Truong) Let ABCD be a square in which the two diagonals intersect at E. A line through A meets BC at M and intersects CD at N. Let K be the intersection point of EM and BN . Prove that CK ⊥ BN. 287. 3 (Nguyen Xuan Hung) Let ABC be a right isosceles triangle, ∠A = 90 ◦ , I be the incenter of the triangle, M be the midpoint of BC. Let MI intersect AB at N and E be the midpoint of IN. Furthermore, F is chosen on side BC such that FC = 3F B. Supp ose that the line EF intersects AB and AC at D and K, respectively. Prove that △ADK is isosceles. 287. 4 (Hoang Hoa Trai) Given a positive integer n, and w is the sum of n first integers. Prove that the equation x 3 + y 3 + z 3 + t 3 = 2w 3 − 1 has infinitely many integer solutions. 10 [...]... Find all functions f : D → D, where D = [1, +∞) such that f (xf (y)) = yf (x) for x, y ∈ D 14 All the best from Vietnamese Problem Solving Journals 295 6 (Nguyen Viet Long) Given an even natural number n, find all polynomials pn (x) of degree n such that i) all the coefficients of pn (x) are elements from the set {0, −1, 1} and pn (0) = 0; ii) there exists a polynomial q(x) with coefficients from the set... AP intersects BC at E and the line BP meets AD at F Prove that the hexagon AM BEQF is cyclic 18 All the best from Vietnamese Problem Solving Journals M B A P F E O′ D Q C 320 3 (Ho Quang Vinh) Let R and r be the circumradius and inradius of triangle ABC; the incircle touches the sides of the triangle at three points which form a triangle of perimeter p Suppose that q is the 1 perimeter of triangle... triangle with AB = AC On the line perpendicular to AC at C, let point D such that points B, D are on different sides of AC Let K be the intersection point of the line perpendicular to AB at B and the line passing through the midpoint M of CD, perpendicular to AD Compare the lengths of KB and KD 22 All the best from Vietnamese Problem Solving Journals 332 3 (Pham Van Hoang) Consider the equation x2 − 2kxy... by p 20 All the best from Vietnamese Problem Solving Journals 328 1 (Bui Van Chi) Find all integer solutions (n, m) of the equation (n + 1)(2n + 1) = 10m2 328 2 (Nguyen Thi Minh) Determine all positive integers n such that the polynomial of n + 1 terms p(x) = x4n + x4(n−1) + · · · + x8 + x4 + 1 is divisible by the polynomial of n + 1 terms q(x) = x2x + x2(n−1) + · · · + x4 + x2 + 1 328 3 (Bui The Hung)... numbers that can not be written in the form ax + by, where x and y are non-negative integers 15 All the best from Vietnamese Problem Solving Journals 297 3 (Le Quoc Han) The circle with center I and radius r touches the sides BC = a, CA = b, and AB = c of triangle ABC at M, N , and P , respectively Let F be the area of triangle ABC and ha , hb , hc be the lengths of the altitudes of △ABC Prove that a)... even All the best from Vietnamese Problem Solving Journals 332 11 (Dang Thanh Hai) Let ABC be an equilateral triangle with centroid O; ℓ is a line perpendicular to the plane (ABC) at O For each point S on ℓ, distinct from O, a pyramid SABC is defined Let φ be the dihedral angle between a lateral face and the base, let γ be the angle between two adjacent lateral faces of the pyramid Prove that the quantity... integer coefficients satisfying the conditions a) p(0) = 1, p(1) = 1; b) p(m) divided by 2003 leaves remainders 0 or 1 for all integers m > 0 26 All the best from Vietnamese Problem Solving Journals 338 4 (Hoang Trong Hao) The Fibonacci sequence (Fn ), n = 1, 2, , is defined by F1 = F2 = 1, Fn+1 = Fn + Fn−1 for n = 2, 3, 4, Show that if a = Fn+1 /Fn for all n = 1, 2, 3, then the sequence (xn ), where... with A = 90◦ Find the locus of points M such that M B 2 − M C 2 = 2M A2 334 7 (Tran Tuan Anh) We are given n distinct positive numbers, n ≥ 4 Prove that it is possible to choose at least two numbers such that their sums and differences do not coincide with any n − 2 others of the given numbers 24 All the best from Vietnamese Problem Solving Journals 335 1 (Vu Tien Viet) Prove that for all triangles ABC... Vietnamese Problem Solving Journals 339 7 (Nguyen Xuan Hung) In the plane, given a circle with center O and radius r Let P be a fixed point inside the circle such that OP = d > 0 The chords AB and CD through P make a fixed angle α, (0◦ < α ≤ 90◦ ) Find the maximum and minimum value of the sum AB + CD when both AB and CD vary, and determine the position of the two chords 340 1 (Pham Hoang Ha) Find the maximum... Find the maximum value of the func√ √ tion f = 4x − x3 + x + x3 for 0 ≤ x ≤ 2 320 2 (Vu Dinh Hoa) Two circles of centers O and O′ intersect at P and Q (see Figure) The common tangent, adjacent to P , of the two circles touches O at A and O′ at B The tangent of circle O at P intersects O′ at C; and the tangent of O′ at P meets the circle O at D Let M be the reflection of P across the midpoint of AB The . times < 5 27 , where there are n radical signs in the expression of the numerator and n −1 ones in the expression of the denominator. 296. 2 (Vi Quoc Dung) Let ABC be a triangle and M the midpoint of BC. The. (Vu Dinh Hoa) The lengths of the four sides of a convex quadri- lateral are natural numbers such that the sum of any three of them is di- visible by the fourth number. Prove that the quadrilateral. such that the sum of the distances from M to AB and CD is equal to the sum of the distances from M to BC and DA. 272. 3 (Ho Quang Vinh) Let M and m be the greatest and smallest numbers in the set